Cognitive OFDM system detection using pilot tones second ...

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Signal Processing

Signal Processing ] (]]]]) ]]]–]]]

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Cognitive OFDM system detection using pilot tones second andthird-order cyclostationarity$

Franc-ois-Xavier Socheleau a,�, Sebastien Houcke a, Philippe Ciblat b, Abdeldjalil Aıssa-El-Bey a

a Institut TELECOM; TELECOM Bretagne; UMR CNRS 3192 Lab-STICC, Universite europeenne de Bretagne, Technopole Brest Iroise-CS 83818,

29238 Brest Cedex, Franceb Institut TELECOM; TELECOM ParisTech, 46 rue Barrault, 75013 Paris, France

a r t i c l e i n f o

Article history:

Received 23 November 2009

Received in revised form

29 March 2010

Accepted 7 July 2010

Keywords:

Cognitive radio

Vertical handoff

Detection

OFDM

Pilot tones

Cyclostationarity

m-Sequence

84/$ - see front matter & 2010 Elsevier B.V. A

016/j.sigpro.2010.07.003

s material is based upon work supported by A

erche under grant ‘‘DEMAIN’’. Part of this wor

RC 2008 [1] and IEEE WCNC 2009 [2].

responding author.

ail addresses: fx.socheleau@telecom-bretagne

Socheleau), sebastien.houcke@telecom-bretag

[email protected] (P. Ciblat),

[email protected] (A. Aıssa

e cite this article as: F.-X. Socheleaur cyclostationarity, Signal Process. (2

a b s t r a c t

The emerging trend to provide users with ubiquitous seamless wireless access leads to

the development of multi-mode terminals able to smartly switch between hetero-

geneous wireless networks. This switching process known as vertical handoff requires

the terminal to first detect the surrounding networks it is compatible with. In the

context where these networks are cognitive, this can be challenging since the carrier

frequency of their access point may change over the time. One solution to overcome this

challenge is to embed network specific signatures in the PHY layer. We here focus on

cognitive OFDM systems and advocate to embed signatures onto pilot tones since (i) it

makes possible to discriminate systems with the same modulation parameters (ii) it

creates easy to intercept signatures implying short detection latency (iii) it avoids

adding any side information dedicated to detection that would reduce systems capacity.

We propose two complementary signature/detection schemes based on second and

third-order statistics, respectively. The first scheme relies on redundancy between pilot

symbols and the second is based on the use of maximum-length sequences. Detailed

numerical examples demonstrate the efficiency of the two detection criteria in realistic

environments.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

To accommodate the anticipated growth in mobileservices, there is ongoing pressure on regulators to makeadditional spectrum available for mobile applications. Asa consequence of the physical limits on the amount ofavailable spectrum, regulatory policies are evolving in thesome bands from the current fixed spectrum rules to

ll rights reserved.

gence Nationale de

k was presented at

.eu

ne.eu (S. Houcke),

-El-Bey).

, et al., Cognitive OFD010), doi:10.1016/j.s

opportunistic spectrum sharing models [3,4]. These newpolicies open up the possibility to make a clever use of theradio resources by developing devices embedding cogni-tive radio (CR) technologies [5,6]. CR enables dynamicspectrum access [7] by sensing the electro-magneticenvironment and specifically by detecting and thenoperating on idle frequency channels (known as whitespaces) at a particular time and place [6]. From a serviceprovider perspective, CR can improve the spectralefficiency [8] and therefore increase the capacity.

In parallel to the development of new mobile services,there is an emerging trend to provide users withubiquitous seamless wireless access. This can be madepossible by taking advantage of the coexistence ofcomplementary heterogeneous networks. In such envir-onment, a challenge is to develop multi-mode terminals(relying on software radio technologies [9]) able to

M system detection using pilot tones second and third-igpro.2010.07.003

www.elsevier.com/locate/sigpro

dx.doi.org/10.1016/j.sigpro.2010.07.003

mailto:[email protected]







Fig. 1. Example of a channel occupation scenario where three systems

share the same frequency range divided into 5 channels. Note that these

channels may not be contiguous.

F.-X. Socheleau et al. / Signal Processing ] (]]]]) ]]]–]]]2

smartly switch from one wireless interface to anotherwhile maintaining IP or voice connectivity and requiredQoS. This switching process is known as ‘‘vertical hand-off’’. In contrast with horizontal handoff (handoff within anetwork made of homogeneous wireless interfaces),vertical handoff triggering is based on a multi-criteriadecision involving signal quality, bandwidth, traffic load,price, battery status, latency, etc. [10,11]. Before measur-ing any of these metrics relevant to trigger a potentialhandoff, the first task to be executed by a multi-modeterminal is to detect the surrounding wireless networks itis compatible with.1

In the classical framework of static frequency alloca-tion policies, the terminal knows the carrier frequencyvalues of the networks of interest. Network access points(AP) detection can thus easily be performed using energydetectors or filters matched to downlink frame preambles.In a cognitive context, AP’s carrier frequency may changeover the time since the spectrum availability depends oninteraction between systems sharing the same band (seeFig. 1). Therefore, if we assume that the spectrum isdivided into a finite number of frequency channels, thenumber and the frequency of the channels used by a givencognitive AP may not be a priori known by the multi-mode terminal. This situation prevents from using energydetectors but still allow the use of matched filters. As areference, for initial detection/synchronization in acognitive context, the IEEE 802.22 working group [12]has defined a superframe structure that encapsulatesclassical frames and that starts with a preambleduplicated on every channel used by an AP. Whensensing all the potential active channels, if a terminalfinds a superframe header on a particular frequencychannel it then obtains all the necessary information(number and frequency of all the channels used by the AP,etc.) to demodulate the frames that follow [13]. Thismechanism is pretty efficient as the channel detection canbe performed by simple cross-correlation between thereceived signal and the known preamble sequence. Inaddition, a terminal only needs to detect a single channel

1 In the remainder of the paper, these wireless networks are

mentioned as ‘‘networks of interest’’.

Please cite this article as: F.-X. Socheleau, et al., Cognitive OFDorder cyclostationarity, Signal Process. (2010), doi:10.1016/j.s

to gather all the information to get connected to thenetwork. However, the drawback of the superframe is thatit decreases the network capacity so that the superframeheader must not be sent frequently to limit the overhead.As an example, the 802.22 superframe header is onlybroadcasted every 160 ms which corresponds to 16frames. In our context where a multi-mode terminal issensing several networks of interest, it may not tolerate atoo long latency in detecting all APs before deciding whichone is the most suited to its needs. For each AP to bedetected, the matched filter approach can indeed be avery long process if the set of possible active channels islarge. For instance, in the worse case scenario, 802.22matched filter detection can take up to 10 s as there canbe more than 60 potential active channels [13].

To limit the handoff latency resulting from matchedfilter detection, an alternative approach consists inembedding network specific patterns or signatures inthe physical signal that are broadcasted at all time.Pattern detection can then be performed at any time bythe multi-mode terminal to identify spectrum bands inuse by the cognitive networks of interest. More generally,the set of minimum requirements that a signature schemeand its associated detectors must meet is2:

�

of a

Migp

Cognitive systems operating in the same frequencybands must be discriminable. This requires to designsignature schemes that can generate as many patternsas operating networks as well as detectors thatguarantee a low false alarm rate.
� System signatures must be quickly detectable. Parallel
sensing of all channels may not be always feasible,specifically in the scenario where the frequency rangeof the possible active channels is wider than thereceiver RF bandwidth of the receiver (see the IEEE802.22 standard [12] for instance). In that case, toavoid long connection latencies, the sensing period ofeach channel must be short. This means that signa-tures must be broadcasted at all time and thatdetection must be effective with small portions ofsignal.
� The signature scheme must not decrease the system
capacity. Signaling overhead must be prohibited toavoid any bandwidth loss.

In contrast with licensed user detection [14], cognitivesystem detection at negative SNR is not required here.Detectors must only guarantee good performance in SNRranges where systems experience bit error rates lowenough to operate. Detection as such is not much ofinterest if the vertical handoff cannot be triggered after-ward due to a poor signal quality.

In this paper we focus on OFDM based systems as it is agood physical layer candidate for cognitive systems since itis scalable by turning on or off some subcarriers and easesspectrum sensing in the frequency domain thanks to itsbuilt-in FFT [15]. Obvious OFDM signatures can be created

2 These requirements are deliberately qualitative to be independent

specific context.

system detection using pilot tones second and third-ro.2010.07.003


F.-X. Socheleau et al. / Signal Processing ] (]]]]) ]]]–]]] 3

thanks to the cyclic prefix (CP) as its duration andperiodicity could be system specific. That is why OFDMdetection has mainly been studied using various correlationor cyclostationary properties induced by the CP [16–20].However, the performance of this approach is drasticallydependent on the cyclic prefix duration and can be affectedby multipath propagation channels. Moreover, such methodis totally inefficient in the presence of a zero-padded OFDMsystem which may be relevant in a cognitive context [21]. Inaddition, in a CR scenario, it is very likely that OFDMsystems competing for the access to the same frequencyband will have the same (or very close) modulationparameters (subcarrier spacing, CP length, etc.). The PHYlayer design is indeed strongly driven by features related tothe spectrum in which a system operate (propagationchannel, available bandwidth, etc.). Therefore, methods thatinvolve more particular signatures in cognitive OFDMsystems may be required and very useful.

In [22,23], the authors suggest approaches usingdedicated subcarriers with cyclostationary patterns. How-ever, dedicating subcarriers to only embed signatures mayhave a cost as it adds overhead and thus reduces networkscapacity. One way to address this issue is to mergesubcarriers used for system detection with existing pilottones typically applied in wireless systems. Pilot tones areof great interest since they are (almost) always present inthe transmitted signals and therefore easy to interceptand avoid adding any signaling overhead dedicated todetection purposes. Existing pilot tones are mainlydesigned for channel estimation [24], synchronization[25] and/or to carry system control information(or signaling) [26–28].

Therefore, adding a new purpose to pilot subcarriers(for instance, system detection) requires dealing withsome already existing constraints, hereafter itemized intotwo categories:

1.

Po

Time–frequency pilot tones distribution—signaturepatterns cannot impose a specific pilot tone arrange-ment in the time–frequency plane as it is usuallydriven by channel estimation requirements [29].

2.
Pilot sequences carrying signaling information—thissecond constraint, usually added on top of the first one,only affects systems that transmit protocol signalingthrough pilot tones. In this case, pilots embedsequences belonging to a finite set of codewordsknown at reception which are decoded by simplecross-correlation (e.g. [27,28]). In this context, pilotsequences have to verify low cross-correlation proper-ties to avoid wrong decoding at reception.
In this contribution, we analyze two solutions to embeddetection signatures onto pilot tones of OFDM signalswithout decreasing networks capacity. In both cases,signature detection is robust to the channel as well as tosynchronization impairments. This detection is only basedupon the knowledge of pilot structures without knowledgeof pilot symbols so that, in contrast with coherent detection,it can be performed on every portion of the received signal.The first approach induces correlation between pilot

lease cite this article as: F.-X. Socheleau, et al., Cognitive OFDrder cyclostationarity, Signal Process. (2010), doi:10.1016/j.s

subcarriers and exploits the deterministic and periodiccharacteristics of pilot mapping in the time–frequencydomain. For this method, system signatures and theirassociated detectors are thus based on second-order cyclos-tationary properties. The second approach considers systemsusing pilots to carry protocol signaling and relies on the useof maximum-length sequence also known as m-sequence.Thanks to their low correlation properties, m-sequences aregood candidates for standard signaling. We also demonstratethat such sequences show higher-order properties relevantto distinguish systems from each other and thereforeadvocate to generalize their use in a cognitive context.

The paper is organized as follows: Section 2 describesthe signal pre-processing implemented in the terminalfront-end prior to detection and also presents the OFDMsystem model. Section 3 recalls the cyclostationarity basisand introduces a novel second-order pilot-induced cy-clostationary (PIC) signature scheme with its correspond-ing detector. In Section 4, m-sequences properties aredetailed, a m-sequence (MS) signature scheme is pro-posed, and the associated detection cost function isderived. Detection performance is assessed throughsimulations in Section 5. Finally, conclusions are pre-sented in Section 6.

2. Signal pre-processing and OFDM system model

2.1. Signal pre-processing

The current design of multi-mode terminals is based onsoftware defined radio technology that enables throughsoftware, dynamic reconfiguration of all protocol stacksincluding the physical layer [30]. A basic front-end of asoftware radio is made of a RF module, an IF downconverterand a ADC followed by a digital front-end. This digital front-end embeds downconverters, filters and sample-rate con-verters, that adapt the bandwidth and the samplingfrequency of the acquired signal to match the PHY featuresof the air-interface to be processed [31]. Multi-modeterminals thus have the ability to perform multirate signalprocessing. The output of the digital front-end is then fedinto the baseband processing engine that detects, demodu-lates, decodes, etc. the signal. Note that a basebandprocessing engine can service multiple RF front-ends, eachof which supports specific air interface standards operatingin a specific frequency band of interest [9]. In this paper, welimit the scope of the baseband processing engine todetection. Fig. 2 represents the functional diagram of amulti-mode terminal in a scenario where it senses afrequency channel in which three different OFDMcognitive networks of interest can potentially operate.Since these networks may have different modulationparameters, upon signal acquisition, the digital front-endprovides three output signals, each of which is tailored tothe PHY features of a particular network. Typically, signal A(resp. B and C) has a sampling frequency that is a multiple ofthe subcarrier spacing of system A (resp. B and C). Thesethree signals are then sent to the detector that makes adecision on the presence or absence of these systems in thesensed frequency channel.



Fig. 2. Block diagram of the multi-mode terminal.

Fig. 3. Examples of pilot tone arrangement: (a) block-type configuration,

(b) comb-type configuration, (c) circular configuration.


2.2. Pilot-assisted OFDM system model

Assuming that in a given frequency channel, atransmitted OFDM symbol consists of N subcarriers, thediscrete-time baseband equivalent transmitted signal isgiven by

xðmÞ ¼

ffiffiffiffiffiEs

N

r½xdðmÞþxtðmÞ�, ð1Þ

where

xdðmÞ ¼Xk2Z

XN�1

n ¼ 0n=2I ðkÞ

akðnÞe2ipðn=NÞðm�D�kðNþDÞÞgðm�kðNþDÞÞ

and

xtðmÞ ¼Xk2Z

Xn2I ðkÞ

bkðnÞe2ipðn=NÞðm�D�kðNþDÞÞgðm�kðNþDÞÞ:

ð2Þ

Es is the signal power and ak(n) are the transmitted datasymbols at n-th subcarrier of k-th OFDM block. These datasymbols are assumed to be independent and identicallydistributed (i.i.d.), D is the CP length; m/gðmÞ is the pulseshaping filter. I ðkÞ denotes the set of pilot subcarrierindexes of the k-th symbol and bk(n) are the pilot symbols.

Let {h(l)}l =0,y, L�1 be the baseband equivalent dis-crete-time channel impulse response of length L. Unlessotherwise stated, the channel is assumed to be timeinvariant. Notice that the impact of channel variations isdiscussed in Section 5. At the output of the digital front-end, the samples of the OFDM signal are then expressed as

yðmÞ ¼ e�ið2peðm�tÞ=NþyÞXL�1

l ¼ 0

hðlÞxðm�l�tÞþZðmÞ, ð3Þ

where e is the carrier frequency offset (normalized by thesubcarrier spacing), y the initial arbitrary carrier phase, tthe timing offset and ZðmÞ a zero mean circularlysymmetric complex-valued white Gaussian noise ofvariance s2 per complex dimension.

As depicted in Fig. 3, three different types of pilot tonearrangements are usually set up in OFDM systems to meetthe channel estimation constraints. The first one is theblock type configuration used under the assumption ofslow fading channel. Pilot tones are in that case mappedonto all subcarriers of OFDM symbols within a specific


period K so that I ðkÞ verifies

I ðkÞ ¼f0, . . . , N�1g if k¼mKðm 2 ZÞ,

| otherwise:

(

The second one, comb-type pilot configuration, isintroduced to satisfy the need for equalizing when thechannel quickly varies. Pilot tones are mapped ontocertain subcarriers of each OFDM symbol such thatI ðkÞ ¼ I where I is any subset of f0, . . . ,N�1g8k. The lastarrangement is a circular configuration where the set ofpilot subcarrier indexes periodically change such thatI ðkþKÞ ¼ I ðkÞ where K 2 Z and I ðkÞa|. Such a scheme isused under the assumption of fast fading channel andpresents the interest of avoiding cases where a given pilotsubcarrier is attenuated by the channel for a periodof several symbols. Note that some OFDM systems(e.g. [32,26]) make a joint use of these arrangements.

3. Second-order pilot-induced cyclostationarysignatures

In this section, we propose to induce joint cyclostatio-narity between pilot tones to create system specific




signatures. The basics on cyclostationarity are first recall inSection 3.1. We present in Section 3.2 the degrees offreedom that define a second-order pilot-induced cylosta-tionary (PIC) signature. A detector tailored to the proposedsignature scheme is then detailed in Section 3.3. To facilitatethe derivation and the understanding of the detector, themethod is first introduced by assuming perfect synchroni-zation at reception. This assumption, purely didactic, is thenrelaxed in Section 3.4 in order to make the proposeddetector robust to synchronization impairments.

3.1. Background on cyclostationarity

Two complex discrete time stochastic processes xk andyk, k 2 Z, are said to exhibit joint second-order almostcyclostationarity in the wide sense if the cross-correlationfunction

Rxyðk,uÞ9E½xky�kþu�,

where * stands for complex conjugation, admits a seriesrepresentation

Rxyðk,uÞ ¼Xa2Axy

RaxyðuÞe

i2pak,

where

RaxyðuÞ9 lim

M-þ1

1

2Mþ1

XMk ¼ �M

Rxyðk,uÞe�i2pak ð4Þ

is the cyclic cross-correlation function (CCCF) and

Axy9fa 2 ½�1=2;1=2½j (u s:t: RaxyðuÞa0g

is a countable set of cycle frequencies a [33]. Note that theCCCF is periodic in a with period 1.

3.2. Pilot-induced cyclostationary (PIC) pattern generation

As illustrated in Section 2.2, the time–frequency pilottones distribution is always deterministic to meet thechannel estimation requirements. For instance, [29]demonstrates that the optimal pilots in the sense ofminimizing the channel mean square error (MSE) arethose that are equispaced and equipowered. As thenumber of pilot tones is finite, the deterministic char-acteristic of the pilot tones distribution can be expressedas I ðkþKÞ ¼ I ðkÞ, K 2 Z for any combination of arrange-ments described in Section 2.2. For the particular case ofcomb-type arrangement, note that K=1. Such a periodicityis a useful property that can be exploited to inducecyclostationarity in OFDM frames through careful choiceof pilot symbols bk(n).

Let ck(n) be the k-th symbol on subcarrier n such that

ckðnÞ ¼bkðnÞ if n 2 I ðkÞ,akðnÞ otherwise:

(ð5Þ

Theorem 1. If the pilot tones are designed such that

bkðpÞ ¼ bkþdðp,qÞ ðqÞeij

with dðp,qÞ 2 Z and j 2 ½�p;p½ then the processes {ck(p)}k

and {ck(q)}k are jointly cyclostationary with Aðp,qÞ9AckðpÞckðqÞ


¼ fðm�bK=2cÞ=K , m 2 f0,1, . . . ,K�1gg where bc stands for

integer flooring.

Proof of Theorem 1 is given in Appendix A.In the framework of OFDM system detection, joint

cyclostationary structure is of great interest to generatesystem specific signatures. Theorem 1 indicates that it ispossible to design a given signature by choosing particularcombinations of p,q,d(p,q) and K. For a system with asubcarrier spacing of 1/N and a cyclic prefix length of D,such a signature S is then defined as

S9fðp,q,dðp,qÞ,KÞjAðp,qÞa|g given N and D: ð6Þ

Fig. 4 shows a schematic illustration of the principle ofOFDM joint cyclostationary pattern generation. Thesignature is S={(0,2,d(0,2)=1, K=3), (6,7,d(6,7)=�2, K=3)},given N=8 and any fixed value of D. From Theorem 1,peaks of energy are then obtained for cycle frequenciesa 2 f� 1

3 ,0, 13g. Note that DVB-T [32], fixed WiMAX [34] or

Wifi [35] pilot structures can be seen as (unintentional)PIC signatures.

The signature scheme allows a very large number ofpossible combinations. However, channel estimationusually requires relatively small K (e.g. K=4 for the DVB-T and K=2 or 9 for the Mobile WiMAX [26]). In addition, tomake the signature usable for detection purposes, d(p,q)

has to be upper-bounded by the observation windowlength necessary to perform the detection (see Section 5).Note also that the term eij in Theorem 1 is here to bringflexibility to PIC structures and to prevent coherentaddition of pilot symbols that would increase the peak-to-average-power ratio (PAPR). This phase shift has noimpact on the detection criterion presented in the sequel.

3.3. Pilot-induced cyclostationary pattern detection

3.3.1. Detection cost function

Thanks to Theorem 1, systems satisfying Eq. (1) and (6)have a periodic cross-correlation function Rcðp,qÞ ðk,dðp,qÞÞ ¼

E½ckðpÞc�kþdðp,qÞ ðqÞ� and can thus be discriminated by

exploiting the cyclic cross-correlation function (CCCF)

Racðp,qÞ ðdðp,qÞÞ9 lim

M-þ1

1

2Mþ1

XMk ¼ �M

E½ckðpÞc�kþdðp,qÞ ðqÞ�e�i2pak:

ð7Þ

We propose to perform the system detection on theevaluation of the CCCF energy at cyclic frequenciesa 2 Aðp,qÞ. In practice the symbols ck(n) are only accessiblevia the observations Yk(n) expressed as

YkðnÞ91ffiffiffiffiNp

XN�1

m ¼ 0

y½kðNþDÞþDþm�e�2ipnm=N :

In our context, the terminal knows the subcarrier spacingvalue as well as the cyclic prefix duration of the networksis trying to detect. These values are usually standardizedand do not change on the flight. In the case of perfectsynchronization (i.e. e¼ 0, t¼ 0 and y¼ 0), the observa-tions simplify to

YkðnÞ ¼HðnÞckðnÞffiffiffiffiffiEs

pþNkðnÞ,



Fig. 4. Schematic illustration of the joint cyclostationary pattern generation. Identical pilot symbols are represented with the same filling colors. The

cyclic cross-correlation function (CCCF) is defined in Eq. (7).


where H(n) and NkðnÞ are, respectively, the channelfrequency response at subcarrier n and the noise atsubcarrier n of the k-th received symbol. Note that thesynchronization impairments are studied in Section 3.4.One can easily check that (Yk(p),Yk(q)) are jointly cyclosta-tionary with a cyclic cross-correlation function expressedas

RaY ðp,qÞ ðuÞ ¼ EsR

acðp,qÞ ðuÞHðpÞH�ðqÞ:

Finally, if we assume that M OFDM symbols areavailable at reception, the CCCF energy of yk(p) and yk(q)is evaluated thanks to the cost function defined as

JPIC9Xðp,qÞ2x

Xa2Aðp,qÞ

jbRa~Yðp,qÞ ðdðp,qÞÞj2

0@ 1A, ð8Þ

where

bRa~Yðp,qÞ ðdðp,qÞÞ ¼

1

M�dðp,qÞ

XM�dðp,qÞ�1

k ¼ 0

~Y kðpÞ ~Y�

kþdðp,qÞ ðqÞe�i2pak ð9Þ

and x¼ fðp,qÞjAðp,qÞa| and dðp,qÞ þKrMg. Note that inorder to get the criterion JPIC less sensitive to the receivedsignal gain, each term Yk(n) in Eq. (9) is normalized so that

~Y kðnÞ ¼YkðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidVar½YðnÞ�

q , ð10Þ

where Var[ � ] denotes the variance and

dVar½YðnÞ� ¼1

M

XM�1

k ¼ 0

jYkðnÞj2: ð11Þ

3.3.2. Decision statistics

Considering that a cognitive network ‘ to be detectedembeds a known PIC signature Sð‘Þ and have a subcarrierspacing of 1=Nð‘Þ and a cyclix prefix length of Dð‘Þ, ourdetection problem can be formulated as a binary hypoth-esis test checking for the presence of cyclostationarity.


Given Nð‘Þ and Dð‘Þ,

H0 : 8ðp;q;dðp;qÞ;KÞ 2 Sð‘Þ; Ra

Y ðp;qÞ ðdðp;qÞÞ ¼ 0,

H1 : for some ðp; q; dðp;qÞ;KÞ 2 Sð‘Þ; RaY ðp,qÞ ðdðp,qÞÞa0:

(ð12Þ

To decide the most likely hypothesis, we propose adetection test constrained by the asymptotic false alarmrate (CFAR) quite similar to [36]. The decision is made bycomparing JPIC to a positive threshold such that

JPIC

H14oH0

L,

with L defined as

F JPIC jH0ðLÞ ¼ 1�Pfa, ð13Þ

where F JPIC jH0is the cumulative distribution function (cdf)

of JPIC when H0 holds and Pfa is the tolerated false alarmprobability.

The energy detector structure of the cost function JPIC

as well as the CFAR test is justified by the limited priorknowledge we have on the received signal. In fact, wecannot derive a detector structure or base our decision onthe statistics of JPIC under H1 as the latter depends onparameters such as the noise power or the propagationchannel that are unknown at reception. Moreover, thedetection is based on the knowledge of the pilot structurewithout the knowledge of pilot symbols so that thedetection can be performed on every portion of thereceived signal. This leads to a short detection latencysince there is no need for frame or superframe synchro-nization in contrast with coherent detection methodsusing known symbol sequences (see IEEE 802.22 [12]).

Since the multi-mode terminal cannot exhaustivelyknow all the type of signals that will be fed into itsdetector (it only knows the features of the systems it iscompatible with), in order to derive the asymptotic




properties of JPIC, the CFAR test is based on the followingassumption:

Assumption 1. Under H0, Yk(n) are assumed to be i.i.d.zero-mean random variables.

Assumption 1 is not very restrictive since the noise andthe vast majority of digital modulated signals that do notmatch the PIC signature to be detected can be modeled asi.i.d. zero-mean random variables. However, for the CFARtest to be efficient, Assumption 1 implies that the numberof possible PIC signatures is large enough to design OFDMsystems satisfying Sð‘Þ

TSðmÞ ¼ |, 8‘am.

In order to find a relevant threshold L to perform thedetection, we hereafter derive the asymptotic statisticalbehavior of JPIC under H0.

Theorem 2. Under hypothesis H0, the cumulative distribu-

tion function of JPIC can be expressed as Laguerre series of

the form

F JPIC jH0ðxÞ ¼

e�x=2o

ð2oÞzþ1

xz

Gðzþ1Þ

XkZ0

k!mk

ðzþ1ÞkLðzÞk

ðzþ1Þx

2on

� �,

8n40 and o40,

with z¼Pðp,qÞ2xcardðAðp,qÞÞ and LðzÞk the k-th generalized

Laguerre polynomial verifying

LðzÞk ðxÞ9Xk

m ¼ 0

kþzk�m

� �ð�xÞm

m!:

GðxÞ9R1

0 tx�1e�t dt and ( � )k denotes the Pochhammer

symbol defined as ðxÞk9GðxþkÞ=GðxÞ. The coefficients mk

satisfy the following recurrent relation:

mk ¼1

k

Xk�1

j ¼ 0

mjgk�j, kZ1

with

m0 ¼ 2ðzþ1Þzþ1 ozþ1

zþ1�nYðp,qÞ2x

onþ zþ1�n2ðM�dðp,qÞÞ

� ��cardðAðp,qÞÞ

and

gj ¼�n

zþ1�n

� �j

þXðp,qÞ2x

cardðAðp,qÞÞnð2ðM�dðp,qÞÞo�1Þ

2ðM�dðp,qÞÞonþzþ1�n

� �j

,

jZ1:

Proof of Theorem 2 is given in Appendix B.Note that the Laguerre expansion depends on n and o

that can be arbitrarily chosen, for more details refer to[37]. The choice of these parameters only have an impacton the convergence speed and uniformity. Moreover, forcomputer implementation, the Laguerre expansion has tobe truncated. The number of terms to consider within theseries can be estimated using the truncation erroranalytical expression given in [37]. For instance, only 15terms are needed to get a proper estimate of the cdf underH0 in the situation illustrated in Fig. 4 by setting M=50,n¼ 3 and oC0:206. Finally, as there are no knownanalytical solutions to invert the cdf of JPIC jH0, thethreshold L is approximated using classical iterativemethods (Newton, etc.).


It is important to note in Theorem 2 that L onlydepends on the PIC signature design and on the observa-tion window length. It is not impacted by environmentalparameters such as signal power, propagation channel,etc. Therefore, L has just to be computed once ‘‘off-line’’and then stored in a look-up table of the multi-modeterminal.

3.4. Discussion on the synchronization and channel impact

on the PIC detector

3.4.1. Effect of synchronization impairments

As stated previously, it is unrealistic to consider themulti-mode terminal perfectly synchronized with thesystems is trying to detect since it cannot know a priorie, t and y (see Eq. (3)). It is straightforward to show thatthe phase shift y has no effect on the cost function JPIC

since the detector is based on the CCCF energy estimation.However, timing missynchronization and/or frequencyoffset deteriorate the observations Yk(n) as inter-symbol(ISI) and inter-carrier (ICI) interferences occur [38–40],and therefore directly impact the cost function JPIC. This isillustrated in Fig. 5 where the value of JPIC is plotted fordifferent frequency and timing offsets using a signalmodel similar to the one depicted in Section 5.2. The SNRis set to 0 dB, M=24 and the channel is frequency selectivewith a root mean square (RMS) delay spread equal to aquarter of the CP length. To guarantee the best detectionperformance, it is required to mitigate the time andfrequency offset. Since the proposed detector does notrely on frame synchronization, the search interval for t islimited to [�0.5(N+D), 0.5(N+D)). For e, the searchinterval is system dependent. In an operational scenariosimilar to the context depicted by the 802.22 wirelessgroup [13], the cognitive networks are frequencychannelized which means that they can only transmiton a finite set of carrier frequencies known by thereceiver. Therefore, when sensing a possible activechannel, the cognitive terminal has only to mitigate thefrequency offset due to the oscillator drift and tothe Doppler effect. Depending on the oscillator precision,the frequency band and the terminal mobility, this offsetis usually within the range of few times the subcarrierspacing.

Since existing blind OFDM synchronization methods[41–43] limit the frequency offset correction to plus orminus half the subcarrier spacing, they cannot be used inour context. In order to guarantee the best performanceand to facilitate the communication link establish-ment that follows detection, we suggest to estimate (e,t)thanks to JPIC. Fig. 5 shows that JPIC is maximum for e¼ 0(no ICI) and t¼ 0 (no ISI) which leads to the followingestimator:

½be,bt� ¼ argmaxðe,tÞ

Jðe,tÞPIC , ð14Þ

where Jðe,tÞPIC is defined as in Eq. (8) by replacing Yk(n) by

Y ðe,tÞk ðnÞ91ffiffiffiffiNp

XN�1

m ¼ 0

y½kðNþDÞþDþmþt�e�2ipðnm=N�eÞ: ð15Þ

Detection and synchronization are thus jointly performed.



–0.50

0.5

–5–2.502.55

100

150

200

250

300

350

400

450

50

500

Fig. 5. Effect of synchronization impairments on the cost function JPIC (SNR=0 dB, M=24, b¼ 0:25D).


As the channel is unknown at reception, the analyticalexpression of the derivative of Jðe,tÞPIC cannot be calculated.The optimization problem of Eq. (14) has therefore to besolved using an algorithm that only requires functionevaluations and not derivatives. There are quite a fewoptimization methods in the literature that address thisproblem. For instance, there is the downhill simplexmethod due to Nelder and Mead [44] but it is not veryefficient in terms of the number of function evaluationsthat it requires. Powell’s method [45] or the Davidon–Fletcher–Powell method with finite differences [46,Chapter 10.9] are almost surely faster. For the simulationsin Section 5, we use Powell’s algorithm which is known asone of the conjugate direction algorithms. Conjugatedirections have the special property that minimizationalong one direction is not ‘‘spoilt’’ by subsequent mini-mization along another, so that interminable cyclingthrough the set of directions can be avoided (see [46,Chapter 10.7] for more details). Experiments show thatthis method only requires 50–100 iterations to convergeover a search window of [�0.5(N+D), 0.5(N+D)) for t and[�10, 10] for e.

The solution of Eq. (14) obviously results in acomputational burden increase. However, in a cognitiveradio context, this additional complexity is not prohibitive

compared to classical preamble matched filter detection.If we still consider the IEEE 802.22 cognitive system as areference, for each sensed channel, the superframepreamble matched filter detection would requireterminals to perform a sliding cross-correlation over80 ms on average if the channel is active andover 160 ms if the channel is inactive. This has to becompared with the PIC detection scheme that onlyrequires a signal duration of a few ms (see Section 5)whatever the state of the channel may be (active orinactive). Consequently, even if the computational costper signal sample of the PIC method is higher than thematched filter detection, since the total number ofsamples to be processed by the PIC method is much


smaller than the other method, the overall PIC complexityis still acceptable.

For instance, in the case of preamble detection basedon a matched filter, the complexity is N� (T-1) multi-plication-accumulations (MAC) for each sensed channel,where T is the number of samples in the observationwindow. For a typical IEEE 802.22 scenario with a 6 MHzactive channel, N=2048 and T=0.08�6.106 (on average)which induces a complexity of approximately 9.8�108

MAC. As for the proposed detector, for each sensedchannel and at each iteration of the Powell algorithm, itperforms M FFT and then correlates the cardðxÞ pilot pairsthat define the signature. The overall complexity istherefore ðM � Nlog2NþcardðxÞ �MÞ � Q , where Q is thetotal number of iterations needed for the Powell algo-rithm to converge. As shown in Section 5, the detectoroffers good performance for M=24, cardðxÞ ¼ 30 andQ=100. This results in a complexity of approximately5.4�107 MAC.

Moreover, we recall that pilot tone-based detectionmethods have the major advantage of drastically reducingthe detection latency relatively to matched filter detec-tion. In addition, in the context described in this paper,requiring synchronization for detection is not unreason-able since at the end the multi-mode terminal will have todo it anyway to measure some metrics useful to decidewhether to trigger a handoff or not [47, Chapter 20].

3.4.2. Effect of frequency-selective channels

The frequency selectivity of the propagation channelimpacts the subcarrier-to-noise-ratio of the pilot tones asthey are weighted by H(n). However, similarly to the termeij in Theorem 1 or the initial phase shift y, as long as thechannel is time-invariant, the phase difference betweenH(p) and H(q) has strictly no effect on the performancesince the detection is based on the CCCF energy. We recallthat the impact of channel fluctuations is discussed inSection 5.




4. m-Sequence signatures

PIC signatures present the major advantage of beingeasy to generate and to adapt to any specific pilot tonestime–frequency distribution. However, such a signaturescheme might not be compatible with cognitive systemsthat would use pilot tones for protocol signaling wherecorrelation between pilot symbols cannot be guaranteed.Indeed, in some cases pilot subcarriers can carry systemcontrol information. For instance, solutions have beensuggested to transmit coded vector mask on pilotsubcarriers for PAPR minimization. In this case, pilottones embed a pseudo-random sequence that is eitherused to inform the receiver of the mapping schemeselected3 at the transmitter side [27], or chosen among aset of predefined sequences as the one that provides thelowest PAPR when combined with the data [28]. Inthe first case, pilot tones carry information whereas inthe second they directly reduce the overall PAPR.

In both cases, the main requirement on thesesequences is their (quasi-) orthogonality to ease thedecoding at reception. The receiver only knows the setof predefined sequences but not the actual transmittedsequence. Decoding is thus performed by exhaustivelytesting all possible sequences for each OFDM symbol.

In this context, following the same progression as theprevious section, we here develop a solution relying onthe use of m-sequence (MS) modulated pilot tones toembed signatures in OFDM signals. MS are widely spreadin wireless systems for operation such as scrambling,spectrum spreading or channel estimation [32,34]. Asdetailed hereafter, MS are of great interest since theyshow specific higher-order statistics relevant for systemdetection and meet the (quasi-)orthogonality or lowcross-correlation requirements.4

This section is organized as follows: the basic proper-ties of MS are first recall in Section 3.1. We then definewhat we call a MS signature in Section 4.2. A detectortailored to the proposed signature scheme is detailed inSection 4.3. For the same reasons as for the PIC detector,perfect synchronization is first implicitly admitted toderive the cost function. Finally, some methods tomitigate the propagation channel and synchronizationimpairments are discussed in Section 4.4.

4.1. Background on m-sequence

A maximum length sequence, commonly calledm-sequence, is a type of pseudo-random binary sequencegenerated using maximal linear feedback shift registers and

3 In the selected mapping (SLM) approach, the input data frame is

multiplied by a finite number of random sequences and whichever

resultant sequence has the lowest PAPR is then selected for transmis-

sion. To enable the receiver to recover the data, a pointer to the

multiplying sequence is transmitted as side information carried by the

pilot tones.4 Note that in the example [27] given previously, the side

information is coded using m-sequences and that m-sequences can also

be shown to be good candidates for the PAPR reduction method

proposed in [28].


modulo 2 addition. A necessary and sufficient condition thata sequence be of maximal length (i.e. sequence of length2p�1 for length-p registers) is that its corresponding

generator polynomial, denoted by PMS, be primitive. Suchsequences satisfy the following properties [48].

Property 1. Let wk be a binary MS of length 2p�1 and

�wk ¼ 1�2wk its BPSK associated sequence, the autocorrela-

tion function of �wk is given by

CðkÞ ¼1

2p�1

X2p�2

i ¼ 0

�wi �wkþ i

¼1 if k¼ 0,

1=ð1�2pÞ otherwise:

(

Property 2. Any binary MS wk generated by length p

registers of polynomial PMS ¼Pp

i ¼ 0 aiXi verifies over GF(2)

wk ¼Xp

i ¼ 1

aiwk�i 3Xp

i ¼ 0

aiwk�i ¼ 0

that can also be written asYi2C

�wk�i ¼ 1

where C¼ fi 2 f0, . . . ,pgjai ¼ 1g.

Property 3. If w(k) is a MS then wðkÞ �wðkþ iÞ is also a MS

8i, with � the addition in GF(2). Considering Property 1, this

leads to

limM-þ1

1

M

XM�1

k ¼ 0

Yi2B

�wk�i

!¼

1

1�2p , 8BaC, ð16Þ

where B is any subset of f0, . . . ,2p�2g.

Property 4. All non-zero codewords have an identical

Hamming distance equal to 2p�1.

These properties show that MS are good candidates forprotocol signaling as they have low cross-correlation values.Moreover, MS show a convolution property (Property 2) thatcan be exploited for system identification.

4.2. m-Sequence (MS) pattern generation

In order to facilitate the identification of OFDM systems,we suggest to generalize the use of MS when protocolsignaling through pilot subcarriers is needed. In this case, foreach k, bk(n) in Eq. (2) is a BPSK pilot symbols sequenceassociated with one MS obtained by the generator poly-nomial PMS. To facilitate the identification and avoid toocomplex detection statistics, we limit PMS to trinomials ofthe form 1+Xl+Xp (p4 l). This limitation has no impact onthe signaling properties since the cross-correlation and theHamming distance only depend on the polynomial degreeand not on the cardinal of C. System signature is thusentirely characterized by [k2ZI ðkÞ and PMS.

4.3. m-Sequence pattern detection

4.3.1. Detection cost function

Our objective is now to propose a cost function todetect MS signatures. Contrary to protocol signaling




decoding that usually consists of testing, for each OFDMsymbol, all possible codewords to decide which one is themost likely, we here propose to limit the complexity byverifying whether the codewords have been generatedby the right PMS. If we consider systems with equispacedpilot subcarriers, from the properties depicted inthe previous subsection it comes that the pilot symbolsbk(n) verify

bkðnÞbkðn�lDÞbkðn�pDÞ ¼ 1, 8n 2 fI ðkÞjnZpDg, ð17Þ

where D denotes the number of subcarriers between pilottones. Thus, for equipowered subcarriers, by analogy withthe PIC criterion, we define

Cak ðl,p,DÞ9

XN�1

n ¼ pD

E½ckðnÞc�kðn�lDÞckðn�pDÞ�e�i2pan:

From Eq. (17) and assuming i.i.d. data symbols

E½ckðnÞc�kðn�lDÞckðn�pDÞ� ¼

XbðN�1�n0ðkÞÞ=Dc

m ¼ p

d½n�mD�n0ðkÞ�,

where d½�� is the Kronecker delta and n0(k) denotes thesubcarrier index of the first pilot on symbol k. It followsthat

Cak ðl,p,DÞ ¼

XbðN�1�n0ðkÞÞ=Dc

n ¼ p

e�i2paðnDþn0ðkÞÞ

¼

sin paD N�1�n0ðkÞ

D

� ��p

� �� sinðpaDÞ e�ipaDðbðN�1�n0ðkÞÞ=Dc�pÞ:

Hence, {ck(n)}n is third-order cyclostationary sinceCa

k ðl,p,DÞ shows peak of energy for a 2 A, with A¼fðm�bD=2cÞ=D, m 2 f0,1, . . . ,D�1gg. The system specificsignature is then defined as

S9fðl,p,DÞjAa|g given N and D: ð18Þ

Consequently, we build a cost function JMS based on third-order statistics defined as

JMS9XM�1

k ¼ 0

Xa2AjbCa

k ðl,p,DÞj2 !

, ð19Þ

where

bCak ðl,p,DÞ9

1

N�pD

XN�1

n ¼ pD

~Y kðnÞ ~Y�

kðn�lDÞ ~Y kðn�pDÞe�i2pan:

Note that only considering equispaced pilot subcar-riers does not limit much our criterion as most systemsverify this property thanks to the result on the optimalpilots location presented in [29]. However, to enlarge thescope of the proposed signature scheme, the cost functionof Eq. (19) can easily be adapted to non-equispaced pilotsubcarriers by considering

~Cak ðl,pÞ9

1

cardðI ðkÞÞX

ni2I ðkÞ

~Y kðniÞ~Y�

kðni�lÞ~Y kðni�pÞ

instead of bCak ðl,p,DÞ and with A¼ f0g. Nevertheless, for the

sake of simplicity, we present derivation and simulationresults associated with Eq. (19).


4.3.2. Decision statistics

We here conduct a hypothesis test similar to Section3.3.2. Considering that a cognitive network ‘ to bedetected embeds a known MS signature Sð‘Þ and givenNð‘Þ and Dð‘Þ, H0 and H1 are defined as

H0 : 8k and ðl; p;DÞ 2 Sð‘Þ; Cak ðl; p;DÞ ¼ 0,

H1 : for some k and ðl; p;DÞ 2 Sð‘Þ, Cak ðl; p;DÞa0:

(ð20Þ

Using the same CFAR test as in Section 3.3.2, we nowhave derive the cdf of JMSjH0 to get the decision threshold.Note that for the same reason as in Section 3.3.2, H0

embodies the vast majority of digital modulated signals.However, in the specific case of H0 where an OFDMsystem has the same parameters N,D,D as the system todetect but with a MS structure of a different polynomialPuMS of degree pu, Property 3 indicates that using a CFARdetector can be a concern. In this case the asymptoticdistribution of JMSjH0 is dependent on pu which is notnecessarily known by the multi-mode terminal. Theterminal only knows the parameters of the systems it iscompatible with. For instance, for cardðI ðkÞÞ-þ1 and inthe purely theoretical scenario where the receiver isperfectly synchronized

Cak ðlu,pu,DÞ ¼

Cak ðl,p,DÞ1�2pu

a0: ð21Þ

Thus, without the knowledge of pu we cannot choose athreshold that guarantees a given Pfa. The only way toovercome this problem is to choose an empirical defaultvalue of the degree pu. For the sake of simplicity, weconsider that pu¼ þ1. Impacts of this assumption onperformance are discussed in Section 5.

Following exactly the same reasoning as in Appendix B,under H0 it comes that the coefficients bCa

k ðl,p,DÞ arezero-mean asymptotic uncorrelated Gaussian variables ofvariance 1=ðN�pDÞ. Then, the characteristic function ofJMSjH0 is expressed as

cJMSjH0ðtÞ ¼ 1�

it

N�pD

� ��DM

which leads to the following theorem.

Theorem 3. Under hypothesis H0, the asymptotic cumula-

tive distribution function of JMSjH0 is expressed as

F JMSjH0ðxÞ ¼

gðDM,ðN�pDÞxÞðDM�1Þ!

,

where gða,xÞ is the incomplete gamma function defined as

gða,xÞ9R x

0 ta�1e�t dt.

The decision threshold can now be computed thanks tothe previous theorem.

4.4. Discussion on the synchronization and channel impacts

on the MS detector

4.4.1. Effect of synchronization impairments

Like the PIC criterion, the cost function JMS is affectedby ISI and ICI interferences resulting from missynchroni-zation. As JMS is built on third-order statistics, theattenuation factor imputable to missynchronization isstronger than for JPIC. Moreover, it can be shown that




timing synchronization errors shift the cyclic frequencieswhich results in additional attenuation of the power ofbCa

k ðl,p,DÞ for a 2 A. Therefore, it is strongly recommendedto estimate e and t.

Similarly to the PIC criterion, JMS is maximum in thecase of perfect synchronization which leads to thefollowing estimator:

½be,bt� ¼ argmaxðe,tÞ

Jðe,tÞMS , ð22Þ

where Jðe,tÞMS is defined as in Eq. (19) by replacing Yk(n) byY ðe,tÞk ðnÞ defined in Eq. (15). Note that contrary to the PICcriterion, JMS only depends on the fractional part of e (seeEq. (19), so that Jðe,tÞMS ¼ Jðeþk,tÞ

MS ,k 2 Z. This means thatexisting blind OFDM synchronization methods [41–43]that usually consider a frequency offset within [�0.5, 0.5)could also be used prior to MS detection (refer to Section 5for more details).

4.4.2. Effect of frequency-selective channels

In contrast with the PIC criterion, the propagationchannel is an important element to consider here as it canhave a strong impact on the MS cost function. Forinstance, in the case of a frequency selective channel wecan have situations where the different H(n) can beconsidered as random variables verifying

E½HðnÞH�ðn�lDÞHðn�pDÞ� ¼ 0) E½bCak ðl,p,DÞ� ¼ 0:

This makes the detection impossible. One way toovercome this problem is to estimate and compensatethe phase shift imputable to the channel. In this case, itcan be shown that the cost function only gets deterioratedby the SNR loss due to propagation. The phase shift can beestimated using Eq. (17). If we write H(n) asHðnÞ ¼ jHðnÞjeiyðnÞ and assume perfect synchronizationand a noiseless channel, Eq. (17) yields the followingrelation for Yðn,l,p,DÞ9yðnÞ�yðn�lDÞþyðn�pDÞ,

Yðn,l,p,DÞ ¼ arg½ ~Y kðnÞ ~Y�

kðn�lDÞ ~Y kðn�pDÞ�, 8n 2 I ðkÞ:ð23Þ

Notice that the equation system (23) is under-determinedsince there are cardðI ðkÞÞ unknown for cardðI ðkÞÞ�p

equations. This is not a concern as any solution will haveexactly the same impact on JMS. Therefore, it only sufficesto choose one possible solution to compensate theperformance loss imputable to the channel phase. Inpractice, Eq. (23) cannot be applied as such since thereceiver is not synchronized. This limitation can beaddressed in two steps. First, we slightly relaxed theperfect synchronization assumption by consideringthat the receiver is still symbol and frequency synchro-nized but not frame synchronized anymore. Thus, thereceiver cannot know I ðkÞ for each symbol k. One way toovercome this lack of knowledge is to exploit the factthat I ðkÞ ¼ I ðkþKÞ and that yðnÞ is invariant in k (underthe time invariant channel assumption). In this


case, Ykðn,l,p,DÞ can be estimated as

bYkðn,l,p,DÞ ¼

arg1

bM=Kc�k

XbM=Kc�1�k

m ¼ 0½k�K þ Kmak

~Y ½k�K þKmðnÞ ~Y�

½k�K þKmðn�lDÞ;� ~Y ½k�K þKmðn�pDÞ

264375,

ð24Þ

where [ � ]K denotes the modulo K operation. Finally, tofully relax the perfect synchronization assumption,bYkðn,l,p,DÞ is jointly computed with the maximization ofJðe,tÞMS in Eq. (22). That is to say that for every tested pairðe,tÞ, Yðn,l,p,DÞ is estimated thanks to Eq. (24).

5. Simulations

5.1. Simulation context

In the following, all the results are averaged over 1000Monte-Carlo runs. We consider 512-subcarrier OFDMsystems inspired from the Mobile WiMAX [26] withD=64, 60 pilot, 360 data, 91 guard and 1 DC subcarrier.The subcarriers are equipowered. The Signal-to-Noise

Ratio (SNR) is defined as SNRðdBÞ ¼ 10log10ðEs=s2Þ. Thesimulated propagation channel is a discrete-timefrequency selective channel {hk(l)}l = 0,y, L with L=D andan exponential decay profile for its non-null component

(i.e. E½jhkðlÞj2� ¼ Ge�l=b for l=0,y, L and G is chosen such

thatPL

l ¼ 0 E½jhkðlÞj2� ¼ 1). Notice that b approximately

corresponds to the root mean square (RMS) delay spread.

We recall that {h(l)}l =0,y, L�1, e, y, t, s2, ak(n) and bk(n) areunknown at reception. The probability of detection isdefined as

Pdet ¼ P½JPIC or MS4LjH1�:

For the simulations, uniformly distributed random eand t are generated with �10rer10 which correspondsto a carrier frequency offset of 710 times the subcarrierspacing and with �0:5ðNþDÞrto0:5ðNþDÞ. The opti-mization problems of Eqs. (14) and (22) are solved usingthe conjugate direction method of Powell that searchesthe maximum of an objective function without calculatingits derivative [45]. In order to keep the false alarm rate tothe expected value, maximizing the cost function impliesto change the detection threshold. In fact, if Powell’salgorithm converges after Q iterations and we assumethat under H0 the cost function values are independentfrom an iteration to another, then the detection thresholdhas to verify

ðF JPIC or MSjH0ðLÞÞQ ¼ 1�Pfa:

The limit the computational complexity, Q is upper-bounded by 100 in all the simulations presented here-after.

5.2. PIC signature detection performance

To evaluate the PIC signature detection performance,we induce joint cyclostationarity on 30 pilot pairs withd(p,q)=2 and q=p+210 for any admissible (p,q).



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pfa

Pde

t

M=12M=24M=48

Fig. 6. PIC criterion—receiver operating characteristic with an increasing length of the observation window (b¼ 0:25D, SNR=�5 dB).

–10 –8 –6 –4 –2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pde

t

fd=0 Hzfd=150 Hzfd=300 Hz

Fig. 7. PIC criterion—effect of Doppler spread on the correct detection probability (M=24, b¼ 0:25D, Pfa=0.02).

5 SNR range where OFDM based systems experience bit-error rates

low enough to operate.


In Fig. 6, we plot the receiver operating characteristics(ROC) for various observation window lengths and for aSNR=�5 dB. The tested observation window lengths havebeen chosen to avoid long sensing period in scenarios wherecognitive networks can operate in several frequencychannels. For instance, 12 WiMAX OFDM symbols onlycorrespond to a duration of 1.23 ms. In Fig. 6, we observethat the detection rate is significantly improved as thenumber of available symbols increases. However, it can beseen that the relative performance improvement diminishesas the observation window gets larger.

Fig. 7 highlights the impact of the frequency-selectivechannel when it becomes time-variant. Time variation is


simulated using Jakes model. Various values of maximumDoppler frequencies fd have been inspected for a falsealarm rate set to 2%. It can be seen that our algorithm isquite robust to Doppler spread. For fd=300 Hz (at 3 GHz,this corresponds to a relative velocity of 108 kph), there isa loss up to 4 dB but the detection performance is stillexcellent in the standard SNR operating range5 for amobile environment.



–10 –8 –6 –4 –2 0 2 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pde

t

Autocorrelation, β=0.1DAutocorrelation, β=0.25DAutocorrelation, β=0.5DPIC, β=0.1DPIC, β=0.25DPIC, β=0.5D

Fig. 8. PIC criterion—comparison between the PIC and the autocorrelation methods (M=24, Pfa=0.02).


As stated in the Introduction, most existing OFDMdetection methods rely on the correlation induced by thecyclic prefix (see [17–19] for instance). In Fig. 8, wecompare the correct detection probability versus SNRbetween the proposed PIC criterion and the standardautocorrelation based method for various channel RMSdelay spread. To compare both methods, we consider thatthe correlation-based detection is correct when

N�d=2r argmaxv2½vmin ,vmax�

XMu�v�1

m ¼ 0

yðmÞy�ðmþvÞ

��rNþd=2,

where Mu¼MðNþDÞ, vmin=32 and vmax=2048 whichcorresponds to searching systems from 32 to 2048subcarriers, and where d is the tolerated error on theestimation. We choose d to be conditioned by the Pfa

under the white Gaussian noise hypothesis such thatd¼ ðvmax�vminÞPfa. As expected, the correlation algorithmcompletely falls down as soon as the channel becomessomewhat selective whereas the PIC method is clearlyrobust to frequency selectivity. Moreover, we recall thatthe PIC criterion can discriminate systems with the sameOFDM modulation parameters whereas the correlationalgorithm cannot.

5.3. MS signature detection performance

To validate the m-sequence signature scheme, equis-paced pilot tones with D¼ 7 are simulated. We apply MS ofpolynomial 1+X2+X5 that generates codewords of length 31satisfying 2p

�1rcardðI ðkÞÞ (p=5, cardðI ðkÞÞ ¼ 60).To set a relevant detection threshold, we first evaluate

the specific case ofH0 where an OFDM system has the sameparameters N,D,D as the system to detect but with a MSstructure of a different polynomial PuMS of degree pu (seeSection 4.3.2). Simulation results (not shown here) indicate


that in a realistic context, like the one depicted in thissection, considering pu¼ þ1 has strictly no impact on theperformance. In fact, whatever the SNR and the testedpolynomial (different from 1+X2+X5) may be, the falsealarm rate obtained by the Monte-Carlo runs is alwaysaround the value set for Pfa and not higher. This finds anexplanation in the synchronization and channel phaseestimation procedure. Without the right polynomial knowl-edge, the receiver is unable to get synchronized and tocompensate the channel phase shift using Eqs. (22) and (24).Therefore, the detection threshold can be set without anyknowledge of pu as it will not impact the performance.

Fig. 9 presents the receiver operating characteristicsfor various observation window lengths and for aSNR=4 dB. Similarly to the PIC criterion, MS performanceis improved in parallel with the increase of the number ofavailable symbols.

Fig. 10 shows the channel Doppler spread effect on theMS criterion. First, it can be seen that the MS criterion is lesseffective than the PIC criterion. Using 3rd order statisticsmakes the detection algorithm more sensitive to noise thana 2nd order based criterion. Moreover, the phase estimatorof Eq. (23) is very sensitive to SNR when only few symbolsare available to average the estimate. However, the MSmethod still performs well for low Doppler in the SNR rangeof interest. We recall that in contrast with licensed userdetection, cognitive system detection is mostly required tohave good performance in SNR ranges where systemsexperience bit error rates low enough to operate. Fig. 10also indicates that the MS criterion is quite robust to Dopplerspread below 150 Hz and that performance degrades up to3 dB for a Doppler of 300 Hz. The performance degradation isonce again amplified by the channel phase estimator basedon the time-invariant channel assumption.

As discussed in Section 4.4, the MS criterion is notaffected by the integer part of the frequency offset e so



–2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pde

t

fd=0 Hzfd=150 Hzfd=300 Hz

Fig. 10. MS criterion—effect of Doppler spread on the correct detection probability (M=24, b¼ 0:25D, fd=0 Hz, Pfa=0.02).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pta

Pde

t

M=12M=24M=48

Fig. 9. MS criterion—receiver operating characteristic with an increasing length of the observation window (b¼ 0:25D, SNR=4 dB).


that existing blind OFDM synchronization algorithms canbe applied prior to detection. By comparing the algorithmsuggested in Eq. (22) with the maximum likelihoodestimator derived in [41], Fig. 11 shows the impact thatthe synchronization method can have on the detectionperformance. It can be seen that using Eq. (22) leadsup to a 4 dB performance gain compared to maximumlikelihood synchronization.

6. Conclusion

In this paper, we have introduced two new signatureschemes to address the OFDM system detection challenge


inherent to the vertical handoff process in a cognitiveradio context. Both methods are based on patternsembedded onto pilot tones which has the main advantageof creating signatures easy to intercept, not adding anysystem overhead and making possible the discriminationof systems with similar modulation parameters. Thesignature schemes presented in this contribution considerexisting pilot constraints in order to be compatible withmost standard specifications.

The first scheme, relying on 2nd order joint cyclosta-tionarity, is applicable to any system that has no specificrequirements on the pilot symbol sequences. Simulationshave demonstrated the detection efficiency of this kind of



–2 0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pde

t

PowellMax. Likelihood

Fig. 11. MS criterion—impact of the synchronization method on the detection performance (M=24, b¼ 0:25D, Pfa=0.02).


pattern and have particularly shown its robustness toharsh propagation environment.

The second signature scheme, based on m-sequences,deals with the constraint of protocol signaling and alsooffers good performance however limited by the necessityto blindly mitigate the phase shift induced by thepropagation channel. Robustness improvement may be atopic to consider in forthcoming work. More specifically,can we find other codes that are compliant with protocolsignaling and system signature requirements while offer-ing better detection performance than m-sequences?

Appendix A. Proof of Theorem 1

The CCCF of ck(p) and ck(q) is defined as

Racðp,qÞ ðuÞ9 lim

M-þ1

1

2Mþ1

XMk ¼ �M

E½ckðpÞc�kþuðqÞ�e

�i2pak:

The data symbols ak(n) being i.i.d. and bkðpÞ ¼

bkþdðp,qÞ ðqÞeij, it follows that

E½ckðpÞc�kþdðp,qÞ ðqÞ� ¼ s2

be�ijX‘2Z

d½k�‘K�k0�,

where d½�� is the Kronecker delta, s2b is the variance of

symbols bk(n) and k0 is the index of the first OFDM symbolembedding pilot tones. Hence,

Racðp,qÞ ðdðp,qÞÞ ¼

s2be�ið2pak0 þjÞ

K

Xm2Z

d a�m

K

h i:

Therefore, for a 2 ½�1=2;1=2½, Racðp,qÞ ðdðp,qÞÞa0 iff

a 2 fðm�bK=2cÞ=K ,m 2 f0,1, . . . ,K�1gg.

Appendix B. Proof of Theorem 2

We first derive the expectation and variance of theCCCF bRa

~Yðp,qÞ ðdðp,qÞÞ defined in Eq. (9).


The expectation of bRa~Yðp,qÞ ðdðp,qÞÞ is given by

E½bRa~Yðp,qÞ ðdðp,qÞÞ� ¼

1

M�dðp,qÞ

XM�dðp,qÞ�1

k ¼ 0

E½ ~Y kðpÞ ~Y�

kþdðp,qÞ ðqÞ�e�i2pak

¼ Ra~Yðp,qÞ ðdðp,qÞÞ:

From Assumption 1, it comes that under H0

E½bRa~Yðp,qÞ ðdðp,qÞÞ� ¼ 0: ð25Þ

To compute the covariance, we introduce the covariancematrix defined as

C¼ E½ðR�EfRgÞðR�EfRgÞH�,

where the superscript H stands for transpose conjugateand

R¼ ½bRa0

~Yðp,qÞ ðdðp,qÞÞ,bRa1

~Yðp,qÞ ðdðp,qÞÞ, . . . ,bRacardðAðp,qÞ Þ�1

~Yðp,qÞ ðdðp,qÞÞ�T :

If we now focus on each element ½C�i,j ¼ E½bRai

~Yðp,qÞ

ðdðp,qÞÞðbRaj

~Yðp,qÞ ðdðp,qÞÞÞ

��, we have

½C�i,j ¼1

ðM�dðp,qÞÞ2

XM�dðp,qÞ�1

ki ,kj ¼ 0

E½ ~Y kiðpÞ ~Y

�

kiþdðp,qÞ ðqÞ ~Y�

kjðpÞ ~Y kjþdðp,qÞ ðqÞ�

�e�i2pðkiai�kjajÞ:

Thanks to Assumption 1, we get

E½ ~Y kiðpÞ ~Y

�


kjðpÞ ~Y kjþdðp,qÞ ðqÞ�

¼ E½ ~Y kiðpÞ ~Y

�

kjðpÞ�E½ ~Y

�

kiþdðp,qÞ ðqÞ ~Y kjþdðp,qÞ ðqÞ�:

This term is different from zero only if ki=kj. Moreover, asshown in Eq. (10), ~Y kðnÞ is expressed as a ratio of tworandom variables. The variance estimator introduced inEq. (11) being consistent, it converges almost surely to aconstant denoted vn so that, thanks to the asymptotictheory developed in [49], ~Y kðnÞ converges in distributionto YðnÞ=

ffiffiffiffiffivnp

. Thus, for ki=kj and a time invariant




propagation channel

E½ ~Y kiðpÞ ~Y

�


kjðpÞ ~Y kjþdðp,qÞ ðqÞ� ¼

E½jYkiðpÞj2�E½jYki

ðqÞj2�

vpvq¼ 1

since vn ¼ E½jYkðnÞj2� and thanks to Assumption 1. There-

fore, the asymptotic covariance is expressed as

½C�i,j ¼1

ðM�dðp,qÞÞ2

XM�dðp,qÞ�1

k ¼ 0

e�i2pkðai�ajÞ

¼sinðpðM�dðp,qÞÞðai�ajÞÞ

ðM�dðp,qÞÞ2sinðpðai�ajÞÞ

e�ipðai�ajÞðM�dðp,qÞ�1Þ: ð26Þ

From Eq. (26) we can deduce that the correlation betweenbRai

~Yðp,qÞ ðdðp,qÞÞ and bRaj

~Yðp,qÞ ðdðp,qÞÞ for aiaaj is upper bounded by

j½C�i,jjr1

ðM�dðp,qÞÞ2sinðpðai�ajÞÞ

r1

ðM�dðp,qÞÞ2sinðp=KÞ

which implies that as long as K5pðM�dðp,qÞÞ2, the cyclic

cross-correlation coefficients are asymptotically mutuallyindependent. Thus, thanks to the central limit theoremand according to Eqs. (25) and (26)

bRa~Yðp,qÞ ðdðp,qÞÞjH0/

D CN 0,1

M�dðp,qÞ

� �,

where /D

indicates the convergence in distribution. Thecyclic cross-correlation coefficients estimate beingasymptotic uncorrelated Gaussian variables, it followsthat ~Y jH0 can be expressed as a sum of weighted centralchi-square variables. We now consider the followingproperties:

Property 5. The characteristic function of a central chi-

square variable X of z degree of freedom is written as

cXðtÞ ¼ ð1�2itÞ�z=2.

Property 6. Let X0,X1,y,Xn�1 being mutually independent

random variables of characteristic functions cX0,cX1

, . . . ,cXn�1

. Then, the characteristic function of Y ¼Pn�1

k ¼ 0 akXk is

defined as

cY ðtÞ ¼Yn�1

k ¼ 0

cXkðaktÞ:

Therefore, from Properties 5 and 6, we finally get

cJPIC jH0ðtÞ ¼

Yðp,qÞ2x

1�it

M�dðp,qÞ

� ��cardðAðp,qÞÞ

:

The inversion of this characteristic function using theresults presented in [37] concludes the proof.

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